When I use a Firefox browser, the code works as it's supposed to when I visit the URL "http://labserver/school/hello.html". However, if I attempt to visit the URL "http://192.168.1.243/school/hello.html" I get the following error messages in the Error Console:

If I modify the "url:" value inside the "hello" function to read 'url: "http://192.158.1.243/school/ajax.txt"' then the behavior is reversed. That is to say that the URL "http://192.168.1.243/school/hello.html" works OK and the URL "http://labserver/school/hello.html" does not. The error messages show up in the Error Console with the various particulars reversed.

Now, it occurred to me while I was writing this that if I made sure that the "server" part of the URL matched the server part specified in the "url: value things would work out fine. But since I've come this far I'd like to know if there's anything in the Apache configuration that would allow for either to work. It has also just occurred to me that it could be something in the way JavaScript handles the URI.

In either case, a gentle "look, dummy, it's like this" would be appreciated.

That is because javascript polices does not allow scripting over different servers and or your sever can't resolve the name to the ip.
So you should use url: "http://192.168.1.243/school/ajax.txt", instead of
[b]url: "http://labserver/school/ajax.txt",